
180
\(\widehat{\phantom{\mathit {HS}}}\) 
 HS
HS*(Y,s)Q

180
\(\widehat{\phantom{\mathit {HS}}}\) 
 HS
HS*(Y,v)* 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM*(Y,s)* 

180
\(\widehat{\phantom{\mathit {HS}}}\) 
 HS
HS*(Y,s)Q ,

where the maps \(\iota _*\)  and \(\pi _*\)  have degree zero, see Chapter 4 of [1]}. This is an exact sequence of \(\mathcal {R}\) -modules where on \(\) HM\(\) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM*\( we have that \) V\( acts as \) U2\( and \) Q\( acts as zero. There are analogue triangles for the \textit {from} and \textit {bar} versions; the latter looks like\begin{center}\begin{tikzpicture}(m) [matrix of math nodes,row sep=1.5em,column sep=1em,minimum width=2em]{\cdot &\cdot &\cdot \\\mathbb {F}&\mathbb {F}& \mathbb {F}\\\mathbb {F}&0&\mathbb {F}\\\mathbb {F}&\mathbb {F}&\mathbb {F}\\};[-stealth](m-2-1) edge node [above]{} (m-2-2)(m-4-2) edge node [above]{} (m-4-3)(m-2-3) edge node [above]{} (m-3-1)(m-3-3) edge node [above]{} (m-4-1);\end{tikzpicture}\end{center}repeated in a \) 4\(-periodic fashion. Here the middle column represents \)HM*\(. In particular, in degrees high enough the Gysin triangle for the \textit {to} groups agrees with this too.{\begin{xmlelement*}{proof}Recall that the degree of element at the bottom of the tower in standard Floer homology \widehat{\mathit {HM}}\end{xmlelement*}\mathit {HM}}\smash{\hspace{1.111pt}{\begin{turn}{180}{\widehat{\phantom{\mathit {HM}}}}\end{turn}} \hspace{-1.111pt}\mathit {HM}}{\vphantom{\widehat{\mathit {HM}}}}{}_*\)  is denoted by \(2\delta \)  (so that \(\delta =-h\)  is the Frøyshov invariant with the opposite sign). Inspecting the map \(\pi _*\)  in the bar version of the Gysin triangle (the arrows on the right side in the diagram above), we obtain by commutativity of the diagrams that the bottom of the \(\alpha \) -tower in \(\) HS\(\) 

180
\(\widehat{\phantom{\mathit {HS}}}\) 
 HS
HS*\( (which is in the image under \) i*\( of the elements in the bottom left and right of the diagram above) lies above of the bottom of the \) U\(-tower in \) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM*\(. Notice that this argument shows that \)\(; on the other hand, there is in general no estimate relating \)\( and \)\(.\) Consider now the portion of the \(\beta \) -tower in degree strictly below \(2\delta \) , denote by \(\mathbf {x}\)  the top degree element among them. Because the \(\alpha \)  tower ends in degrees \(\ge 2\delta \) , all the elements in this portion are killed by \(Q\) , hence are in the image of \(\pi _*\) . Choose \(\mathbf {y}\)  in \(\) HM\(\) 

180
\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM*\( such that \) *(y)=x\(; notice that \) y\( does not belong to the \) U\(-tower for degree (parity) reasons. If there are \) k\( elements in this portion of the \)\(-tower, \) Vk-1x0\( hence by commutativity of the diagram also\begin{equation*}\pi _*(U^{2k-2}\cdot \mathbf {y})=V^{k-1}\cdot \mathbf {x}\ne 0,\end{equation*}so that \) U2k-2y0\( and \) t(Y,s)2k-1\(. Hence\begin{equation*}\delta (Y,\mathfrak {s})-\min \lbrace \delta (Y,\mathfrak {s}),\beta (Y,\mathfrak {s})\rbrace \le 2\cdot \#\lbrace \text{elements in $\beta $ tower of degree $<2\delta $}\rbrace =t(Y,\mathfrak {s})+1.\end{equation*}Now, the Gysin triangle for the bar groups shows that the elements of the \)\(-tower are never in the image of the \) U\(-tower; the argument above can then be applied to the part of the \)\(-tower in degrees below \) 2+1\( to obtain the bound\begin{equation*}\beta (Y,\mathfrak {s})-\gamma (Y,\mathfrak {s})\le 2\cdot \#\lbrace \text{elements in $\gamma $ tower of degree $<2\beta +1$}\rbrace =t(Y,\mathfrak {s})+1.\end{equation*}Putting the two estimates together, we obtain{\begin{@align*}{1}{-1}\delta (Y)-\gamma (Y)&=\delta (Y)-\beta (Y)+\beta (Y)-\gamma (Y)\le \\&\le \delta (Y)-\min \lbrace \delta (Y),\beta (Y)\rbrace +\beta (Y)-\gamma (Y)\le \\&\le 2t(Y)+2.\end{@align*}}Finally, applying the inequality to \) Y\(, we also obtain using \) (Y,s)=-(Y,s)\( that\begin{equation*}\alpha (Y)-\delta (Y)\le =2t({Y})+2,\end{equation*}(where we also used \) t(Y,s)=t(Y,s)\( by Poincaré duality), and the result follows.\) We use this to prove the following result, which is the main tool we will use to study the homology cobordism group in the present paper.
